Chaos - James Gleick [66]
The points trace a trajectory that provides a way of visualizing the continuous long-term behavior of a dynamical system. A repeating loop represents a system that repeats itself at regular intervals forever.
If the repeating behavior is stable, as in a pendulum clock, then the system returns to this orbit after small perturbations. In phase space, trajectories near the orbit are drawn into it; the orbit is an attractor.
An attractor can be a single point. For a pendulum steadily losing energy to friction, all trajectories spiral inward toward a point that represents a steady state—in this case, the steady state of no motion at all.
A PHYSICIST HAD GOOD REASON to dislike a model that found so little clarity in nature. Using the nonlinear equations of fluid motion, the world’s fastest supercomputers were incapable of accurately tracking a turbulent flow of even a cubic centimeter for more than a few seconds. The blame for this was certainly nature’s more than Landau’s, but even so the Landau picture went against the grain. Absent any knowledge, a physicist might be permitted to suspect that some principle was evading discovery. The great quantum theorist Richard P. Feynman expressed this feeling. “It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?”
Like so many of those who began studying chaos, David Ruelle suspected that the visible patterns in turbulent flow—self-entangled stream lines, spiral vortices, whorls that rise before the eye and vanish again—must reflect patterns explained by laws not yet discovered. In his mind, the dissipation of energy in a turbulent flow must still lead to a kind of contraction of the phase space, a pull toward an attractor. Certainly the attractor would not be a fixed point, because the flow would never come to rest. Energy was pouring into the system as well as draining out. What other kind of attractor could it be? According to dogma, only one other kind existed, a periodic attractor, or limit cycle—an orbit that attracted all other nearby orbits. If a pendulum gains energy from a spring while it loses it through friction—that is, if the pendulum is driven as well as damped—a stable orbit may be the closed loop in phase space that represents the regular swinging motion of a grandfather clock. No matter where the pendulum starts, it will settle into that one orbit. Or will it? For some initial conditions—those with the lowest energy—the pendulum will still settle to a stop, so the system actually has two attractors, one a closed loop and the other a fixed point. Each attractor has its “basin,” just as two nearby rivers have their own watershed regions.
In the short term any point in phase space can stand for a possible behavior of the dynamical system. In the long term the only possible behaviors are the attractors themselves. Other kinds of motion are transient. By definition, attractors had the important property of stability—in a real system, where moving parts are subject to bumps and jiggles from real-world noise, motion tends to return to the attractor. A bump may shove a trajectory away for a brief time, but the resulting transient motions die out. Even if the cat knocks into it, a pendulum clock does not switch to a sixty-two–second minute. Turbulence in a fluid was a behavior of a different order, never producing any single rhythm to the exclusion of others. A well-known characteristic of turbulence was that the whole broad spectrum of possible cycles was present at once. Turbulence is like white noise, or static.